Direct investigation of anisotropic suspension structure in pressure-driven flow

C. Gao,1 S. D. Kulkarni,2 J. F. Morris,2 and J. F. Gilchrist1,*1Department of Chemical Engineering, Lehigh University, Bethlehem, Pennsylvania 18015, USA

2Benjamin Levich Institute and Department of Chemical Engineering, The City College of New York, New York, New York 10031, USA�Received 20 July 2009; published 15 April 2010�

Evidence is presented to show the microstructural anisotropy responsible for normal stress in shearedsuspensions. Particle velocimetry is combined with three-dimensional particle locations obtained via confocalmicroscopy at rest. A range of volume fractions � and local shear rates �̇ are investigated in a weaklyBrownian pressure-driven suspension. At high �̇, the pairwise distribution shows a strong probability along theaxis of compression similar to observations from Stokesian dynamics simulation at �=0.32. At the channelcenter where �̇→0, the concentrated suspension at �=0.56 behaves as a confined isotropic fluid.

DOI: 10.1103/PhysRevE.81.041403 PACS number�s�: 82.70.Kj, 47.57.J�, 47.57.E�

The shearing of suspensions with moderate to high par-ticle volume fractions is known to generate normal stresses�1–3�. This force primarily acts orthogonal to the direction offlow and is generally believed to be a result of multiparticlehydrodynamic interactions. The normal stresses may be in-terpreted as the tendency to dilate under deformation; whenconstrained to a fixed volume, dilation does not occur but thenormal stresses are measurable. Rheological measurement ofthese normal stresses is challenging because the normalstresses are small until approaching jamming at high solidvolume fractions; however, recent macroscopic measure-ments in Couette flow have succeeded in measuring osmoticpressure �4�. In the presence of nonlinear shear fields, thesenormal stresses result in the migration of particles �5�, whichdemixes suspensions. The development of constitutive equa-tions to describe this behavior in a wide range of flows hasbeen slow due to the fact that these normal stresses are theresult of subtle structural anisotropies. Detailed three-dimensional �3D� particle-level information from a single ex-periment is necessary in order to identify local structure. Thedevelopment of a fundamental understanding of these multi-particle hydrodynamic interactions will impact a suspensionprocess design for mixing, separations, and transport andwill facilitate a deeper understanding of natural processesincluding sediment transport and blood flow.

Brownian and non-Brownian systems ranging in volumefraction 0.3���0.5 have been studied extensively throughtheory and simulations to identify and understand the effectof shear-induced microstructural anisotropies on the develop-ment of normal stresses. Stokesian dynamics simulations�6,7� predict a higher correlation of nearest-neighbor interac-tions along the compressive axes of the flow in the v−�vprojection of the pair distribution function, g�r�. From anisotropic state, these dynamics develop quickly with the on-set of flow �8�. While the normal stresses disappear upon thecessation of flow in non-Brownian systems, the structuralanisotropy is preserved �8,9�. This anisotropy is destroyedupon the reversal of the flow direction and re-emerges ori-ented relative to the new flow direction �8�. This connectionbetween local structural anisotropy and discontinuities ob-

served upon flow reversal was confirmed by direct imagingof experiments exhibiting this fore-aft asymmetry �10� butexperiments to date lack the detail available via simulation.For ��0.5, shear deformation results in a higher degree oflocal order including particle chaining and crystallization�11�.

We present measurements of 3D particle-level anisotropygenerated from shear in a pressure-driven flow using confo-cal laser scanning microscopy �CLSM�. The advantage ofthis technique is that direct 3D imaging gives data that canbe directly compared to simulation and probed using a mul-titude of analyses. We used monosized suspensions of 2a=1.01 �m diameter SiO2 microspheres at a bulk volumefraction �bulk=0.41 in 3:1 glycerol:water with 0.1 mMRhodamine-B fluorescent index-matching solution. To mini-mize electrostatic interactions in order to predominantlystudy the result of hydrodynamic interactions, we addedNaOH and NaCl to adjust the pH to 8.0 and the screeninglength to �−1=1.5 nm. We introduce this suspension into a50 mm�40 �m�100 �m microchannel at a pressuredrop as high as 1 atm using compressed nitrogen. The coor-dinates �x, y, and z� that represent the �v, �v, and ��v�directions are reported as dimensionless quantities scaled bythe 20 �m half-height of the channel in the �v direction.The nonlinear shear gradient as a result of pressure-drivenflow induces significant suspension migration. The finalsteady volume fraction profile � far from the entrance,shown in Fig. 1�a�, is higher at the center and lower near thewalls. The local viscosity depends on the local volume frac-tion; therefore, the velocity profile deviates from axial Poi-seuille flow of a Newtonian fluid. Fully developed far fromthe entrance region, this generates a velocity profile acrossthe y direction at the center having a roughly 1 mm/s centervelocity, plotted in Fig. 1�b�. For comparison axial Poiseuilleflow of a Newtonian fluid, as calculated using the solutionderived in �11�, is also shown in Fig. 1�b�. The local shearrate �̇, calculated from the local slope of the velocity profileover measured values within the nearest 6 �m of each point,is nearly linear and is used to define the local Péclet number,Pe�60�̇a3 / �kT�, where 0 is the fluid viscosity and kT isthe thermal energy. The nonlinear velocity profile as a resultof pressure-driven flow creates shear rate variation in the zdirection as well; however, as we are considering only the*gilchrist@lehigh.edu

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small central portion �20% of the overall z width�, the shearrate deviation in this direction can be neglected.

Dynamic confocal laser scanning microscopy �VTeye,Visitech Int.� allows single plane particle tracking for veloci-metry to measure velocity profiles as well as measurement ofparticle concentration patterns in the plane transverse to thedirection of flow �12,13�; however, its scanning speed is in-sufficient to track fast moving particles in 3D space and en-able investigations of the microstructure. To eliminate tran-sients, after flowing for 10 min we quickly stop the flow,similar to �14�, and scan the entire depth of the channel in the�v direction within a region of interest �ROI� located at themidpoint at the center of the ��v direction �20�40�20 �m3�. The use of compressed nitrogen gives a stabledriving pressure and the fast pressure release using a three-way valve near the channel entrance allows a rapid arrest ofsuspension flow without flow reversal. In non-Browniandensity-matched systems, the structure that produces normalstresses is preserved upon the cessation of the flow providedthe flow does not reverse �8,9�. After suspension flow stops,the ROI is scanned at high speed of �10 pixels /�m over2.4 s and the 3D locations of particles within are identified�15,16�. Fast scanning allows identification of particle loca-tions before significant Brownian diffusion and sedimenta-tion alter the local structure. Multiple scans are obtained andthe insignificant Brownian and buoyant drift that occurs inthe time between scans suggests that the structure is unal-tered in the duration of a single scan. The average particlenumber density n̄=N /V is used to calculate the local volumefraction as �= n̄ 4

3r3; this provides accurate particle densitymeasurements �Fig. 2�a�� that match well with measurementstaken during the flow.

With the particle locations, local coordination can be used

to investigate the suspension structure. A representative 3Dreconstruction of the particles found in the ROI is shown inFig. 2�b�. The particles are size and color coded to distin-guish disordered particles �1/5th size and gray� from thosehaving local face-centered-cubic �FCC� or hexagonal-close-packed �HCP� structure or neighbors thereof �red and ren-dered to scale�, as defined in previous investigation of thestructure of liquids by identifying bond order parametersbased on bond angle weighted averages �17�. Most orderedparticles are located near the center, where they compriseroughly 20% of particles in the region −0.25�y /H�0.25.The generation of crystalline order in coexistence with a liq-uid structure is expected for an equilibrium dispersion �i.e.,Pe=0� for 0.494���0.545.

Using the accelerated Stokesian dynamics technique �18�with Brownian motion �19�, we simulate the motion of col-loidal particles suspended in sheared viscous fluid in thelimit of zero particle inertia where viscous effects are domi-nant. For a system of N particles suspended in an incom-pressible Newtonian fluid, fluid motion is governed by theStokes equations while particle motion is described by theN-body Langevin equation; this equation is integrated to ob-tain time evolution of the particle positions, as detailed inRefs. �18,19�. The simulations are performed using periodicboundary conditions in all directions with Lees-Edwardsboundary conditions for shear flow �20�. Results reportedhere use N=512 particles in the simulated unit cell. The bulksimple shear flow is u= �ux ,uy ,uz�= ��̇y ,0 ,0�, so that x-y isthe plane of shear and y-z is the plane perpendicular to themean flow.

The particle configurations, obtained experimentally andin simulations by varying the sampling location within thechannel and Pe, respectively, are used to determine the pairdistribution function, g�r�= P1�1�r� / n̄. P1�1�r� is the probabil-ity of finding a particle at position r given a particle centeredat the origin. To compare with the experimental data, g�r� isevaluated in different planes. For example, g�x ,y� definesg�r� within the shear plane. To calculate g�x ,y� we considerall pair separations in which the magnitude of the z compo-nent of the separation satisfies rz�2a. Experimentally, thedata are averaged over five experiments. Simulations averagea sufficiently large number of configurations that the resultbecomes independent of the number of configurations �7,21�.The pair distribution g�x ,y� is evaluated from a histogram of

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FIG. 1. �Color online� �a� The particle volume fraction profiletransverse to the direction of flow and the region of interest wheredata are sampled after stopping the flow. �b� The measured velocityprofile in the region of interest �triangles� and the parabolic profile�green dotted line� for comparison. The shear rate profile calculatedfrom the velocity in y axis �diamonds� fit with a linear trend line�red solid line�.

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FIG. 2. �Color online� �a� Density profile across the ROI from3D particle locations shows the result of migration. �b� Renderingof experimentally obtained particle locations as small gray pointsand true to scale larger red particles for those particles having FCCor HCP bond order.

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pair separation vectors in the bins normalized by the respec-tive bin volume ��r=0.2a and � =3.6°�. The distributionsg�z ,y� and g�x ,z� are evaluated in the same fashion. All plotsof g�r� span dark blue to red colors representing the range0�g�r��2.

In Fig. 3, the pair distribution functions measured andcalculated at Pe=1700 are seen to have a central region ofparticle exclusion. As a result of the way the data are repre-sented, symmetry through the origin is mandatory in all pairdistribution functions. For g�x ,y� representing the shear-plane structure, inside r�2a a strong pair correlation existsin the compressional region near the line x=−y and less inthe extensional region neighboring a particle. This appears atr�2a due to the way the out-of-plane data are projected. Inthe simulation, the near-contact peak is similar to that foundin previous studies �6,7� and is the primary microstructuralgenerator of macroscopic normal stresses. This peak deviatesfrom the experimental measurement in that the correlation isless uniform in the direction and, in particular, is strongeralong the x and y axes. The shape of the wake is different inthe extensional region, and there is a stronger correlation

along the x axis in the experimental results. In both experi-ments and simulations, a second peak just inside r=4a isapparent but a direct comparison of the structural details isdifficult due to the weakness of correlation in this region andthe resulting lower signal-to-noise ratio.

In g�x ,z� for the microstructure in the velocity-vorticityplane, the fore-aft symmetry apparent in the simulations isnot found experimentally. We hypothesize that the shear gra-dient along the z direction, �̇z, in the experiment is nonzeroeven though measurements are near z=0 and are significantenough to create similar increased correlation along the com-pressional region. In the first ring of higher probability insider=2a, simulations show lower correlation in the x directionnear z=0 that results from averaging the g�x ,y� data. Thissame projection for the experimental data results in higher

yvs.x(∇vvs.v)

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FIG. 4. �Color online� Experimental compilation of local pairdistribution functions g�x ,y� �first row�, g�x ,z� �second row�, andg�z ,y� �third row� located at −0.1�y /H�0.1, �local=56.4%, Pe=0.

yvs.x(∇vvs.v)

experiment simulation-4a -2a 0 2a 4a-4a -2a 0 2a 4a

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FIG. 3. �Color online� Experimental compilation �left� andStokesian dynamics �right� simulations of local pair distributionfunctions. Pair distribution functions g�x ,y� �first row�, g�x ,z� �sec-ond row�, and g�z ,y� �third row� located at −0.85�y /H�−0.65,�=31.9%, Pe=1700 in the channel and simulated at Pe=1700.

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pairwise probability in the x direction near z=0 and lowerprobability along the x=y and x=−y directions. A secondring of higher probability is found again inside r=4a for bothexperiments and simulations. In the vorticity-shear gradientplane where flow goes into the plane of the page for theexperiment, there are first- and second-order rings of higherprobability in the pair distribution function g�z ,y� as foundin the other two planes. There are also dissimilarities, wheresimulations predict near uniformity in probability in the an-gular direction, the experiments show slightly higher prob-ability in the shear plane in the vorticity direction.

The pair distribution functions are also reported for thecenter of the channel −0.1�y /H�0.1 where the local vol-ume fraction peaks at �=0.564 �Fig. 4�. At this volume frac-tion the suspension is expected to generate crystalline orderat equilibrium. Likewise, crystallization has been shown inthe presence of shear at similar volume fractions �8,22–25�.At this point in the channel, although the concentration gra-dients driven by shear migration are assumed to be well es-tablished, particles near the center only spend roughly 30 s inthe microchannel; it is unlikely that the suspension hasreached equilibrium in this weakly Brownian suspension.This, plus consideration of any residual shear in either the yor z direction, may be the reason why only a fraction ofparticles near the center of the channel rendered in Fig. 2 arefound to have local order. The velocity-vorticity plane g�x ,z�displays three clear rings of higher pairwise probability justinside r=2a, r=4a, and r=6a, and arguably a fourth near thecorners at r=8a. In the nearest-neighbor ring, there is somevariation in probability in the angular direction with the

highest probability along the axes and the lowest along x=y and x=−y. The observations in the pairwise distributionsin the alternate planes g�x ,y� and g�z ,y� show little correla-tion outside the x-z plane with the exception of higher prob-ability in the nearest-neighbor pair in g�x ,y� along the y axis.This anisotropy suggests that if crystallization does occur, itwill be nucleated and propagate in the x-z plane.

In conclusion, CLSM-enabled 3D measurement of par-ticle locations in experiments enables a direct microstructurecomparison to particle locations generated by Stokesian dy-namics simulations. The apparent anisotropies lead to shearmigration in nonlinear velocity profiles that enable probingof various Pe’s and �’s in a single experiment. Experimentaland simulated pair distribution functions largely agree, al-though the details deviate possibly due to the lateral confine-ment of the suspension in the z direction. This flow-arrestingtechnique can be expanded to probe the structure of anyhydrodynamic-dominated nonkinetic or weakly kinetic sus-pension to probe heterogeneities that result from polydisper-sity, particle shape anisotropy, and some non-Newtonian flu-ids comprising the continuous phase in a wide range offlows.

The authors want to thank J. Brady for insightful conver-sations and E. Weeks for the 3D particle tracking algorithms.The microfabrication was performed at the Cornell NanoS-cale Facility. This material is based upon work supported bythe National Science Foundation under Grant No. 0630191�J.F.G.�, funding from the ACS Petroleum Research Fund�J.F.G. and J.F.M.�, and North American Mixing Forum�J.F.G.�.

�1� J. J. Stickel and R. L. Powell, Annu. Rev. Fluid Mech. 37, 129�2005�.

�2� J. F. Morris and F. Boulay, J. Rheol. 43, 1213 �1999�.�3� I. E. Zarraga, D. A. Hill, and D. T. Leighton, J. Rheol. 44, 185

�2000�.�4� A. Deboeuf, G. Gauthier, J. Martin, Y. Yurkovetsky, and J. F.

Morris, Phys. Rev. Lett. 102, 108301 �2009�.�5� C. J. Koh, P. Hookham, and L. G. Leal, J. Fluid Mech. 266, 1

�1994�.�6� D. R. Foss and J. F. Brady, J. Fluid Mech. 407, 167 �2000�.�7� J. F. Morris and B. Katyal, Phys. Fluids 14, 1920 �2002�.�8� J. J. Stickel, R. J. Phillips, and R. L. Powell, J. Rheol. 51,

1271 �2007�.�9� V. G. Kolli, E. J. Pollauf, and F. Gadala-Maria, J. Rheol. 46,

321 �2002�.�10� F. Parsi and F. Gadala-Maria, J. Rheol. 31, 725 �1987�.�11� R. J. Butera, M. S. Wolfe, J. Bender, and N. J. Wagner, Phys.

Rev. Lett. 77, 2117 �1996�.�12� C. Gao, B. Xu, and J. F. Gilchrist, Phys. Rev. E 79, 036311

�2009�.�13� C. Gao and J. F. Gilchrist, Phys. Rev. E 77, 025301�R� �2008�.

�14� D. Dendukuri, S. S. Gu, D. C. Pregibon, T. A. Hatton, and P. S.Doyle, Lab Chip 7, 818 �2007�.

�15� Using IDL code developed by Eric Weeks at Emory Universityusing algorithms developed by Crocker and Grier �16�.

�16� J. C. Crocker and D. G. Grier, J. Colloid Interface Sci. 179,298 �1996�.

�17� P. J. Steinhardt, D. R. Nelson, and M. Ronchetti, Phys. Rev. B28, 784 �1983�.

�18� A. Sierou and J. F. Brady, J. Fluid Mech. 448, 115 �2001�.�19� A. J. Banchio and J. F. Brady, J. Chem. Phys. 118, 10323

�2003�.�20� M. Allen and D. Tildesley, Computer Simulation of Liquids

�Oxford University Press, New York, 1987�.�21� S. D. Kulkarni and J. F. Morris, J. Rheol. 53, 417 �2009�.�22� R. L. Hoffman, J. Rheol. 16, 155 �1972�.�23� L. B. Chen, M. K. Chow, B. J. Ackerson, and C. F. Zukoski,

Langmuir 10, 2817 �1994�.�24� R. M. Amos, J. G. Rarity, P. R. Tapster, T. J. Shepherd, and S.

C. Kitson, Phys. Rev. E 61, 2929 �2000�.�25� P. Holmqvist, M. P. Lettinga, J. Buitenhuis, and J. K. G. Dhont,

Langmuir 21, 10976 �2005�.

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